Existence and examples of quantum isometry group for a class of compact metric spaces

Abstract

We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banica's definition for finite metric spaces. For metric spaces $(X,d)$ which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on $(X,d)$. In fact, our existence theorem applies to a larger class, namely for any compact metric space $(X,d)$ which admits a one-to-one continuous map $f : X \raro \IR^n$ for some $n$ such that $d_0(f(x),f(y))=\varphi (d(x,y))$ (where $d_0$ is the Euclidean metric) for some homeomorphism $\varphi$ of $\IR^+$. As concrete examples, we obtain Wang's quantum permutation group $\cls_n^+$ and also the free wreath product of $\IZ_2$ by $\cls_n^+$ as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in \cite{huang1}.

To appear in Adv. Math

References in corpus (2)

• Faithful compact quantum group actions on connected compact metrizable spaces
• Invariant subsets under compact quantum group actions

Cited by in corpus (4)

• Liberations and twists of real and complex spheres
• Quantum Isometry group of dual of finitely generated discrete groups and quantum groups
• Quantum isometry group of dual of finitely generated discrete groups- $\text{II}$
• Topological automorphism groups of compact quantum groups